Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.
A)0 hours
B)3.61 hours
C)693.38 hours
D)5.61 hours
E)8.80 hours

Question

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t > 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.
A)0 hours
B)3.61 hours
C)693.38 hours
D)5.61 hours
E)8.80 hours

Quizplus · Accepted Answer

To find the t-value at which x is maximum, we need to find the derivative of $x(t)$ with respect to $t$ and set it equal to zero to solve for $t$. The derivative of $x(t) = \frac{5000t}{t^2 + 13}$ is found using the quotient rule: $x'(t) = \frac{5000(t^2 + 13) - 5000t(2t)}{(t^2 + 13)^2}$. Simplifying this gives $x'(t) = \frac{5000(13 - t^2)}{(t^2 + 13)^2}$. Setting $x'(t) = 0$ gives $13 - t^2 = 0$, which leads to $t^2 = 13$. Taking the square root of both sides gives $t = \sqrt{13}$, which is approximately 3.61. Therefore, the t-value at which x is maximum is approximately 3.61 hours.

Medication x(t)=5000tt2+13x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }x(t)=t2+135000t​

Medication $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$